Question

(overline{de}) is a perpendicular bisector of (overline{fg}), and they intersect at point (h). which of the following statements must be true? select the two correct answers. (1 point)
(square overline{dh}congoverline{eh})
(square overline{fh}congoverline{gh})
(square overline{fg}congoverline{de})
(square angle dhfcongangle dhg)
(square angle fdhcongangle dgh)

Explanation:

Step1: Recall the definition of a perpendicular bisector

A perpendicular bisector of a segment divides the segment into two equal parts. Since \(DE\) is the perpendicular bisector of \(FG\) and they intersect at \(H\), by the definition of a bisector, \(FH = GH\), so \(\overline{FH}\cong\overline{GH}\).

Step2: Analyze the angle - related property

Since \(DE\perp FG\) at \(H\), \(\angle DHF\) and \(\angle DHG\) are right angles. Right angles are congruent, so \(\angle DHF\cong\angle DHG\) (by the right - angle congruence theorem: all right angles are congruent).

Answer:

\(\overline{FH}\cong\overline{GH}\), \(\angle DHF\cong\angle DHG\)